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1: 4.10 Integrals
4.10.1 d z z = ln z ,
4.10.2 ln z d z = z ln z - z ,
4.10.4 d z z ln z = ln ( ln z ) ,
4.10.7 0 x d t ln t = li ( x ) , x > 1 .
The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). …
2: 4.2 Definitions
The principal value, or principal branch, is defined by … We regard this as the closed definition of the principal value. … The principal value is … Another example of a principal value is provided by …
3: 4.13 Lambert W -Function
We call the solution for which W ( x ) W ( - 1 / e ) the principal branch and denote it by Wp ( x ) . …
See accompanying text
Figure 4.13.1: Branches Wp ( x ) and Wm ( x ) of the Lambert W -function. … Magnify
4.13.5 Wp ( x ) = n = 1 ( - 1 ) n - 1 n n - 2 ( n - 1 ) ! x n , | x | < 1 e .
4.13.10 Wp ( x ) = ξ - ln ξ + ln ξ ξ + ( ln ξ ) 2 2 ξ 2 - ln ξ ξ 2 + O ( ( ln ξ ) 3 ξ 3 ) ,
4.13.11 Wm ( x ) = - η - ln η - ln η η - ( ln η ) 2 2 η 2 - ln η η 2 + O ( ( ln η ) 3 η 3 ) ,
4: 6.15 Sums
6.15.2 n = 1 si ( π n ) n = 1 2 π ( ln π - 1 ) ,
6.15.3 n = 1 ( - 1 ) n Ci ( 2 π n ) = 1 - ln 2 - 1 2 γ ,
6.15.4 n = 1 ( - 1 ) n si ( 2 π n ) n = π ( 3 2 ln 2 - 1 ) .
5: Stephen M. Watt
He was one of the original authors of the Maple and Axiom computer algebra systems, principal architect of the Aldor programming language and its compiler at IBM Research, and co-author of the MathML and InkML W3C standards. …
6: Bonita V. Saunders
She is the Visualization Editor and principal designer of graphs and visualizations for the DLMF. … As the principal developer of graphics for the DLMF, she has collaborated with other NIST mathematicians, computer scientists, and student interns to produce informative graphs and dynamic interactive visualizations of elementary and higher mathematical functions over both simply and multiply connected domains. …
7: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
The principal value of the exponential integral E 1 ( z ) is defined by … Unless indicated otherwise, it is assumed throughout the DLMF that E 1 ( z ) assumes its principal value. … This is the principal value; compare (6.2.1). …
8: 4.37 Inverse Hyperbolic Functions
§4.37(ii) Principal Values
Compare the principal value of the logarithm (§4.2(i)). … The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by … Graphs of the principal values for real arguments are given in §4.29. … Throughout this subsection all quantities assume their principal values. …
9: 4.1 Special Notation
k , m , n integers.
10: 4.6 Power Series
4.6.1 ln ( 1 + z ) = z - 1 2 z 2 + 1 3 z 3 - , | z | 1 , z - 1 ,
4.6.2 ln z = ( z - 1 z ) + 1 2 ( z - 1 z ) 2 + 1 3 ( z - 1 z ) 3 + , z 1 2 ,
4.6.3 ln z = ( z - 1 ) - 1 2 ( z - 1 ) 2 + 1 3 ( z - 1 ) 3 - , | z - 1 | 1 , z 0 ,
4.6.4 ln z = 2 ( ( z - 1 z + 1 ) + 1 3 ( z - 1 z + 1 ) 3 + 1 5 ( z - 1 z + 1 ) 5 + ) , z 0 , z 0 ,
4.6.5 ln ( z + 1 z - 1 ) = 2 ( 1 z + 1 3 z 3 + 1 5 z 5 + ) , | z | 1 , z ± 1 ,